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Can anyone explain how Rgd is obtained? I plugged both Ix into Rgd expression but did not get the correct result.

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here is my attempt

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I cannot proceed. What did I do wrong? How can I get rid of Vx ?

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You are nearly done.

$$ R_{gd} = \frac{V_xR'_L}{g_mV_{gs}R'_L-I_xR'+V_x} $$

Use \$V_{gs}=-I_xR'\$:

$$ ...=\frac{V_xR'_L}{-g_mR'_LI_xR'-I_xR'+V_x}\cdot\frac{\frac{1}{V_x}}{\frac{1}{V_x}}=...\\ $$ note that those pesky \$\frac{I_x}{V_x}=\frac{1}{R_{gd}}\$. You need to rearrange your equation and get to:

$$ R_{gd}\left(1-\frac{R'}{R_{gd}}-\frac{gmR'_LR'}{R_{gd}}\right)=R'_L $$

Getting your book result is now straightforward.

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